Thing, even if I change the base_path value in config/gitlab.yml, it keeps trying to look up the same folder.

Only possible directions for the solution, that I found on google, indicated, that it is somehow related to SElinux. The suggested command was: find . -print0 |xargs -0 -n 1 sudo setfattr -h -x security.selinux. But i have no clue where to execute it and it tells me that command setfattr not found .. or something. Also, I kinda doubt, that unchanging configuration might too be affected by SElinux.

So this is your real issue here (before starting tweaking the folder permissions): gitlab somehow ignores your config/gitlab.yml setting.
Double-check for any typo in it, and make sure it is taken into account at all (by removing it for instance, to see if gitlab complains about that file missing).